Published paper
Abstract
The paper establishes foundational switching, covering-graph, matroid, incidence-matrix, and vector-representation results for signed graphs. Theorem 8B.1 is the rank and linear-dependence input used by the later signed-graph geometry survey and orientation paper.
Role in dependence graphs
Proof-critical source
Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements
This paper is included only for the following marked statement:
- Theorem 8B.1 · printed p. 70 in the two-up journal reprint; supporting Lemma 8A.3 on p. 69Rank and dependence of the signed incidence/vector representation used in the later arrangement identity and the first orientation proof.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact author-hosted journal reprint · Discrete Applied Mathematics 4 (1982), 47–74
Thomas Zaslavsky. Signed graphs. Discrete Applied Mathematics 4 (1982), 47–74.
Open audited source ↗01Statements3 reported findingsContains wrong statements
Most foundational signed-graph statements, including the marked vector-representation Theorem 8B.1, are correct. The exact 1982 article is not fully correct as published: the author's correction record identifies errors in Theorem 5.1 and Corollary 7D.3(g), with a later correction also required for part (f). Those results are outside the dependency branch used here.
Theorem 8B.1Correct and unaffected by known corrections
The marked signed-vector dependence theorem is correct
Printed journal page 70 · Section 8B and Theorem 8B.1
The signed incidence vectors have rank equal to the number of vertices minus the number of balanced connected components, and their minimal dependencies are exactly the signed-graphic circuits described earlier. Switching multiplies vertex coordinates by signs and therefore preserves dependence. This is the precise branch used by the 1991 orientation proof and the 2012 geometry survey.
Exact author-hosted journal reprint ↗Theorem 5.1 and Corollary 7D.3Incorrect as published · published corrections exist
The original article has documented corrected assertions
Printed journal pages 57–58 and 65–66 · Theorem 5.1 and Corollary 7D.3; correction record linked from the author's publication page
The author's publication record explicitly identifies the 1983 erratum as correcting Theorem 5.1 and Corollary 7D.3(g), and records that part (f) was corrected in later work. Accordingly, the uncorrected 1982 text cannot receive a globally correct statement rating. The dependency trace does not rely on any of these clauses; it uses only Theorem 8B.1.
Sections 1–4 and 6–8BCorrect in the checked branch
The remaining structural signed-graph claims are consistent
Printed journal pages 47–56 and 58–70 · switching, balance, coverings, matroids, and vector representations
Switching equivalence, balance by cycle signs, the signed covering graph, and the frame-matroid circuit types agree under restriction and contraction. The vector representation realizes exactly those circuits. No additional contradiction was found in the portions recursively needed for the marked source claims.
02Proofs2 reported findingsContains incorrect or incomplete proofs
The proof of the marked Theorem 8B.1 is correct, but the article as a whole contains proofs whose published conclusions required an erratum and a later correction. Because the staged reprint is the uncorrected 1982 text, those branches remain incorrect as printed even though the exact dependency branch is verified.
Proof of Theorem 8B.1Correct
The marked theorem follows from the preceding vector-rank lemma
Printed journal pages 69–70 · Lemma 8A.3 and proof of Theorem 8B.1
The printed proof defines the dependence matroid of the incidence-vector map, invokes the determinant characterization in Lemma 8A.3, and says the remaining details are routine. Together those steps establish the representation claim; the article does not present the spanning-forest elimination argument previously described here.
Corrected branchesIncorrect or incomplete as printed
The uncorrected proofs cannot support their original full conclusions
Printed journal pages 57–58 and 65–66 · proofs surrounding Theorem 5.1 and Corollary 7D.3; 1983 erratum and later correction record
The existence of itemized corrections means the original derivations do not establish the corresponding clauses exactly as printed. This audit records the source-level defect rather than silently substituting a later theorem. None of those clauses feeds Theorem 8B.1, so the marked recursive edge remains verified.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.